--> pdiv_inc
Polynomial division with increasing terms powers
SYNTAX
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pdiv_inc // Displays this help
[q, r] = pdiv_inc(p, d)
[q, r] = pdiv_inc(f)
PARAMETERS
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p,d,q,r: Single polynomials with real or complex coefficients.
p : polynomial to be divided
d : divisor. Its first non-null coefficient of lowest power must
be of power equal or lower than p's one.
q : quotient, with deg(q) <= deg(p)
r : remainder, with deg(r) <= deg(p)+deg(d)
such that: p = q*d + r. p and d are not necessarily coprime.
f : Single rational, with coprime polynomials p = numer(f) and d = denom(f)
DESCRIPTION
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pdiv_inc(p, d) performs the Euclidian division p/d with polynomial terms
of p and d sorted by increasing powers. It returns the quotient q and the
remainder r, such that p = q*d + r.
Detailled example:
2 - x + 3x^2 | 1 + 2x^2
---------------------------|------------
2 + 4x^2 | 2 - x - x^2
- x - x^2 |
- x - 2x^3 |
- x^2 + 2x^3 |
- x^2 - 2x^4 |
2x^3 + 2x^4 |
Hence, (2 - x + 3x^2) = (2 - x - x^2)*(1 + 2x^2) + (2x^3 + 2x^4)
NOTES:
* If d==0, q=0 and r=p are returned whatever is p.
* if both p and d are limited to a decimal, there are set to p+0*x and d+0*x
WARNING: If f is specified explicitly through a p/d ratio with p and d not
coprime, their common factors are first simplified, and the p = q*d + r
relationshift holds only with simplified p => sp and d => sd.
Example: p = poly([1 2 2 3 3],'x'), d = poly([2 3 4],'x'), p/d
[q,r] = pdiv_inc(p/d), p-q*d-r, f = p/d; numer(f)-q*denom(f)-r
REFERENCE
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Comments, scoring and bug reports are welcome on
http://fileexchange.scilab.org/toolboxes/449000#new_comment
SEE ALSO
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pdiv : Polynomial division by decreasing terms powers
modulo : Remainder after polynomial division
ldiv : Limited Taylor series of a polynomial ratio
EXAMPLES
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x = poly(0,"x");
// 1) Simple example
p = 1 + 3*x + 2*x^2 - 7*x^3
d = -1 + x - 2*x^2
[q,r] = pdiv_inc(p,d), p-q*d-r // => 0
// 2) With some non-leading null coefficients in p and/or d
p = 1 + x + 3*x^3
d = 1 + 2*x^2
[q,r] = pdiv_inc(p,d), p-q*d-r // => 0
// 3) With p not starting with x^0
p = x - x^2 + 2*x^3
d = -1 + 2*x^2
[q,r] = pdiv_inc(p,d), p-q*d-r // => 0
// 4) With p and d not starting with x^0
p = x^2 + 2*x^3
d = -x + 2*x^2 - x^3
[q,r] = pdiv_inc(p,d), p-q*d-r // => 0
// 5) with complex coefficients:
p = -1 + (1+%i)*x - %i*x^2 ;
d = %i - x + (1-%i)*x^2 ;
[q,r] = pdiv_inc(p,d);
sci2exp(q), sci2exp(r), sci2exp(p-q*d-r) // => 0
// 6) d must start at a degree lower or equal to p's starting degree
p = 2 - x + 3*x^2
d = 2*x - x^2
[q,r] = pdiv_inc(p,d) //=> Error
// 7) Random example
p = poly(grand(1,5,"uin",-9,9), "x", "coeff")
d = poly(grand(1,7,"uin",-9,9), "x", "coeff")
[q,r] = pdiv_inc(p,d), clean(p-q*d-r) // => 0